Recurrence procedure for calculating kernels of the nonlinear collision integral of the Boltzmann equation

Abstract

A recurrence procedure for a sequential construction of kernels \(G_{{l_1},{l_2}}^l\) (c, c₁, c₂) appearing upon the expansion of a nonlinear collision integral of the Boltzmann equation in spherical harmonics is developed. The starting kernel for this procedure is kernel G_0,0⁰ (c, c₁, c₂) of the collision integral for the distribution function isotropic with respect to the velocities. Using the recurrence procedure, a set of kernels \(G_{{l_1},{l_2}}^{ + l}\) (c, c₁, c₂) for a gas consisting of hard spheres and Maxwellian molecules is constructed. It is shown that the resultant kernels exhibit similarity and symmetry properties and satisfy the relations following from the conservation laws.